A multidimensional central limit theorem with speed of convergence for axiom a diffeomorphisms

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Abstract

Let T: X → X be an Axiom A diffeomorphism, m the Gibbs state for a Hölder continuous function g. Assume that f: X → Rd is a Hölder continuous function with ∫X fdm = 0. If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ2 :=σ2(f) such that Snf√n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ2. Moreover, there exists a real number A > 0 such that, for any integer n ≥, wherem*(1/√n Snf) denotes the distribution of1nSnf with respect to m, and II is the Prokhorov metric. © 2011 Wuhan Institute of Physics and Mathematics.

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Xia, H., & Tan, D. (2011). A multidimensional central limit theorem with speed of convergence for axiom a diffeomorphisms. Acta Mathematica Scientia, 31(3), 1123–1132. https://doi.org/10.1016/S0252-9602(11)60303-2

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